Automorphic Forms on Gl

By Jacquet, Herve

Book Id: WPLBN0000696823
Format Type: PDF eBook
File Size: 1.60 MB
Reproduction Date: 2005

Title:	Automorphic Forms on Gl
Author:	Jacquet, Herve
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Language:	English
Subject:	Science., Mathematics, Logic
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Citation APA MLA Chicago Jacquet, H. (n.d.). Automorphic Forms on Gl. Retrieved from http://gutenberg.us/

Description
Mathematics document containing theorems and formulas.

Excerpt
Excerpt: Two of the best known of Hecke?s achievements are his theory of L-functions with grossencharakter, which are Dirichlet series which can be represented by Euler products, and his theory of the Euler products, associated to automorphic forms on GL(2). Since a grossencharakter is an automorphic form on GL(1) one is tempted to ask if the Euler products associated to automorphic forms on GL(2) play a role in the theory of numbers similar to that played by the L-functions with grossencharakter. In particular do they bear the same relation to the Artin L-functions associated to two-dimensional representations of a Galois group as the Hecke L-functions bear to the Artin L-functions associated to one-dimensional representations? Although we cannot answer the question definitively one of the principal purposes of these notes is to provide some evidence that the answer is affirmative. The evidence is presented in S12. It come from reexamining, along lines suggested by a recent paper of Weil, the original work of Hecke. Anything novel in our reexamination comes from our point of view which is the theory of group representations. Unfortunately the facts which we need from the representation theory of GL(2) do not seem to be in the literature so we have to review, in Chapter I, the representation theory of GL(2, F) when F is a local field. S7 is an exceptional paragraph. It is not used in the Hecke theory but in the chapter on automorphic forms and quaternion algebras. Chapter I is long and tedious but there is nothing hard in it. Nonetheless it is necessary and anyone who really wants to understand L-functions should take at least the results seriously for they are very suggestive.

Table of Contents
Table of Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Chapter I: Local Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 ? 1. Weil representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 ? 2. Representations of GL(2, F) in the non-archimedean case . . . . . . . . . . . 15 ? 3. The principal series for non-archimedean fields . . . . . . . . . . . . . . . . 58 ? 4. Examples of absolutely cuspidal representations . . . . . . . . . . . . . . . . 77 ? 5. Representations of GL(2,R) . . . . . . . . . . . . . . . . . . . . . . . . 96 ? 6. Representation of GL(2,C) . . . . . . . . . . . . . . . . . . . . . . . . . 138 ? 7. Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 ? 8. Odds and ends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Chapter II: Global Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 ? 9. The global Hecke algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 189 ?10. Automorphic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 ?11. Hecke theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 ?12. Some extraordinary representations . . . . . . . . . . . . . . . . . . . . . 251 Chapter III: Quaternion Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 267 ?13. Zeta-functions forM(2, F) . . . . . . . . . . . . . . . . . . . . . . . . . 267 ?14. Automorphic forms and quaternion algebras . . . . . . . . . . . . . . . . . 294 ?15. Some orthogonality relations . . . . . . . . . . . . . . . . . . . . . . . . 304 ?16. An application of the Selberg trace formula . . . . . . . . . . . . . . . . . . 320